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Questions and Answers
Which of the following expressions is equivalent to $sin(3\alpha)$?
Which of the following expressions is equivalent to $sin(3\alpha)$?
- $3sin(\alpha) + 4sin^3(\alpha)$
- $4sin(\alpha) - 3sin^3(\alpha)$
- $3sin(\alpha) - 4sin^3(\alpha)$ (correct)
- $4sin(\alpha) + 3sin^3(\alpha)$
Given $cos(\alpha) = \frac{1}{4}$, find the value of $cos(3\alpha)$.
Given $cos(\alpha) = \frac{1}{4}$, find the value of $cos(3\alpha)$.
- $-\frac{47}{64}$ (correct)
- $\frac{47}{64}$
- $-\frac{11}{16}$
- $\frac{11}{16}$
Simplify the expression: $2sin(x)cos(3x)$
Simplify the expression: $2sin(x)cos(3x)$
- $sin(4x) + sin(2x)$
- $cos(4x) - cos(2x)$
- $sin(4x) - sin(2x)$ (correct)
- $cos(4x) + cos(2x)$
If $tan(\alpha) = 2$ and $tan(\beta) = \frac{1}{3}$, find the value of $tan(\alpha + \beta)$.
If $tan(\alpha) = 2$ and $tan(\beta) = \frac{1}{3}$, find the value of $tan(\alpha + \beta)$.
Given $sin(\alpha) = \frac{3}{5}$ and $cos(\beta) = \frac{5}{13}$, where both $\alpha$ and $\beta$ are in the first quadrant, find the value of $cos(\alpha - \beta)$.
Given $sin(\alpha) = \frac{3}{5}$ and $cos(\beta) = \frac{5}{13}$, where both $\alpha$ and $\beta$ are in the first quadrant, find the value of $cos(\alpha - \beta)$.
Which of the following is equivalent to $cos(2x)$?
Which of the following is equivalent to $cos(2x)$?
What is the simplified form of the expression $\frac{1 - cos(2\theta)}{sin(2\theta)}$?
What is the simplified form of the expression $\frac{1 - cos(2\theta)}{sin(2\theta)}$?
Determine the expression equivalent to $sin(x + \frac{\pi}{2})$.
Determine the expression equivalent to $sin(x + \frac{\pi}{2})$.
If $tan(\theta) = -\frac{3}{4}$ and $\theta$ is in the second quadrant, find the value of $sin(2\theta)$.
If $tan(\theta) = -\frac{3}{4}$ and $\theta$ is in the second quadrant, find the value of $sin(2\theta)$.
Simplify: $cos(\frac{\pi}{2} - \theta)$
Simplify: $cos(\frac{\pi}{2} - \theta)$
Flashcards
sin(α + β) Identity
sin(α + β) Identity
sin(α + β) = sinα cosβ + cosα sinβ
cos(α + β) Identity
cos(α + β) Identity
cos(α + β) = cosα cosβ - sinα sinβ
tan(α + β) Identity
tan(α + β) Identity
tan(α + β) = (tanα + tanβ) / (1 - tanα tanβ)
sin(2α) Identity
sin(2α) Identity
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cos(2α) Identity
cos(2α) Identity
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tan(2α) Identity
tan(2α) Identity
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sin(3α) Identity
sin(3α) Identity
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cos(3α) Identity
cos(3α) Identity
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tan(3α) Identity
tan(3α) Identity
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sin(-θ) Identity
sin(-θ) Identity
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Study Notes
- Trigonometry study notes
Sum/Differences identities
- sin(α+β) = sinα cosβ + cosα sinβ
- sin(α-β) = sinα cosβ - cosα sinβ
- cos(α+β) = cosα cosβ - sinα sinβ
- cos(α-β) = cosα cosβ + sinα sinβ
- tan(α+β) = (tan α + tan β) / (1 - tanα tan β)
- tan(α-β) = (tanα - tan β) / (1 + tanα tan β)
- cot(α+β) = (cotα cotβ - 1) / (cot β+ cot α)
- cot(α-β) = (cotα cotβ + 1) / (cot β - cot α)
Double angle identities
- cos 2α = cos² α - sin² α
- cos 2α = 1 - 2sin² α which implies 2sin² α = 1 - cos² α
- cos 2α = 2cos² α - 1, and 1 + cos2α = 2 cos² α
- sin²(α/2) = (1 - cosα) / 2
- tan 2α = (2 tan α) / (1 - tan² α)
- cos 2α = (1 - tan² α) / (1 + tan² α)
- sin 2α = (2 tan α) / (1 + tan² α)
Triple angle identities
- sin3α = 3sinα - 4sin³ α
- cos3α = 4cos³α - 3 cosα
- tan3α = (3 tanα - tan³ α) / (1 - 3 tan² α)
Product to sum identities
- 2sinα cosβ = sin(α+β) + sin(α-β)
- 2cosα sinβ = sin(α+β) - sin(α-β)
- 2cosα cosβ = cos(α+β) + cos(α-β)
- 2sinα sinβ = cos(α-β) - cos(α+β)
Sum to product identities
- sinα + sinβ = 2sin((α+β)/2) cos((α-β)/2)
- sinα - sinβ = 2cos((α+β)/2) sin((α-β)/2)
- cosα + cosβ = 2cos((α+β)/2) cos((α-β)/2)
- cosα - cosβ = -2cos((α+β)/2) sin((α-β)/2)
Other identities
- sin² α - sin² β = sin(α+β)·sin(α-β)
- cos² α - cos² β = sin(β+α)·sin(β-α)
Even/Odd identities
- sin(-θ) = -sinθ
- cos(-θ) = cos θ
- tan(-θ) = -tan θ
- cosec(-θ) = -cosec θ
- sec(-θ) = sec θ
- cot(-θ) = -cot θ
Confunction identities
- sin(π/2 - θ) = cos θ
- cosec(π/2 - θ) = sec θ
- tan(π/2 - θ) = cot θ
- cot(π/2 - θ) = tan θ
- sec(π/2 - θ) = cosec θ
- cot(π/2 - θ) = tan θ
Half Angle identities
- sin(θ/2) = ±√((1 - cos θ)/2)
- cos(θ/2) = ±√((1 + cos θ)/2)
- tan(θ/2) = ±√((1 - cos θ)/(1 + cos θ))
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